
UQLAB user manual
1.5 Sampling random vectors
In most uncertainty quantification applications, sampling the input vectors is as important
as properly represent them. However, most of the existing random sampling strategies (e.g.
Monte Carlo sampling, latin hypercube sampling (LHS), pseudorandom sequences, etc.) pro-
duce samples in the unit hypercube distributed according to X ∼ U([0, 1]
M
). Amongst the
many strategies available to generate samples distributed according to a specific PDF F
X
,
UQLAB approaches the problem in terms of isoprobabilistic transforms. An isoprobabilistic
transform is a map of the form (Lebrun and Dutfoy, 2009):
X = T (U ) s.t. X ∼ F
X
, U ∼ F
U
(1.16)
or, in other words, it is a change of variables that transforms a sample of random vector
U ∼ F
U
into a sample of random vector X ∼ F
X
. Of particular interest for sampling
purposes is the transform between the unit hypercube U ∼ U([0, 1]
M
), and any other random
vector with joint distribution F
X
.
In the following, isoprobabilistic transforms will be derived for the purposes of sampling
independent random vectors as well as random vectors with Gaussian copula.
1.5.1 Isoprobabilistic transform of independent marginals
Consider a sampling of size N of the unit hypercube Z =
u
(1)
, . . . , u
(N)
∼ U([0, 1]
M
).
Due to the independence between the components of the random vector X ∼ F
X
, a simple
isoprobabilistic transform can be used to transform Z into X =
x
(1)
, . . . , x
(N)
∼ F
X
:
x
(i)
j
= F
−1
X
j
(u
(i)
j
) (1.17)
where F
−1
X
j
denotes the inverse CDF of the j-th marginal of the random vector X.
1.5.2 Generalized Nataf transform
In the case of dependent variables, several extra steps are needed to properly transform
a sample from the unit hypercube to the desired joint PDF. A powerful tool is given by the
generalized Nataf transform (Lebrun and Dutfoy, 2009). Given Sklar’s theorem in Eq. (1.10),
it follows that that a sample Z from a random vector X ∼ F
X
(x) can be obtained from
a sample of the underlying copula C =
c
(1)
, . . . , c
(N)
with a component-by-component
isoprobabilistic transform similar to that in Eq. (1.17):
x
(i)
j
= F
−1
X
j
(c
(i)
j
) (1.18)
where F
−1
X
j
denotes the inverse CDF of the j-th marginal of the random vector X. Moreover,
the underlying copula distribution is a multivariate Gaussian. An important property of the
Gaussian copula is that it is elliptical, which means that an isoprobabilistic transform exist
that maps samples from an elliptical copula with correlation matrix R to the same copula
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